Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros.
Degree​ 4;  ​ zeros: 3 minus 4 i ; minus 2 multiplicity 2

To construct a polynomial with real coefficients, we need to use the given information:  

- **Degree 4**: The polynomial will have 4 roots (including multiplicities).
- **Zeros**:
  - \(3 - 4i\): Since the coefficients must be real, the complex conjugate \(3 + 4i\) is also a zero.
  - \(-2\) with **multiplicity 2**: This means the factor \((x + 2)\) appears twice.

### Step-by-step construction:

1. **Form factors for the zeros:**
   - For the complex conjugate pair \(3 - 4i\) and \(3 + 4i\):
     \[
     \left( x - (3 - 4i) \right) \left( x - (3 + 4i) \right)
     \]
     Use the difference of squares formula:
     \[
     \left( x - 3 \right)^2 + 16 = x^2 - 6x + 25
     \]

   - For the zero \(-2\) with multiplicity 2:
     \[
     (x + 2)^2 = x^2 + 4x + 4
     \]

2. **Form the polynomial**:
   Multiply the two factors:
   \[
   f(x) = \left( x^2 - 6x + 25 \right) \left( x^2 + 4x + 4 \right)
   \]

3. **Expand the product**:
   Use distribution (multiplying every term in one factor by every term in the other):

   \[
   f(x) = x^4 + 4x^3 + 4x^2 - 6x^3 - 24x^2 - 24x + 25x^2 + 100x + 100
   \]

   Now, combine like terms:
   \[
   f(x) = x^4 - 2x^3 + 5x^2 + 76x + 100
   \]

Thus, the polynomial is:
\[
f(x) = x^4 - 2x^3 + 5x^2 + 76x + 100
\]

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Do you have any questions or need more details?  

Here are 5 related questions:  
1. How do you find the complex conjugate of a given complex number?  
2. What is the difference of squares formula and when is it used?  
3. How do multiplicities affect the shape of a polynomial’s graph?  
4. What is the fundamental theorem of algebra?  
5. How do you determine the degree of a polynomial?  

**Tip:** Multiplying complex conjugate pairs always gives a quadratic expression with real coefficients.

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros: 3 - 4i ; -2 multiplicity 2.

Learn how to form a degree 4 polynomial with real coefficients given the zeros 3 - 4i and -2 (multiplicity 2). This solution covers polynomial construction, complex conjugates, and multiplicity of roots. Ideal for students in Grades 10-12.

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